Uni"ld States
Department of
Agriculture
Forest Service
Pacific Southwest
Forest and Range
Exprlment Station
P.O. Box 245
Berkeley
California 94701
Research Note
PSW-378
Romain M. Mees
December I985
A
Mees, Romain M. Sitnrrlating initial attack
with I W O fire containment models. Res.
Note PS W-378. Berkeley, CA: Pacific Southwest Forest and Range Experiment Station,
Forest Service, U.S. Department of Agriculture; 1985. 7 p.
Given a variable rate of fireline construction
and an elliptical fire growth model, two methods for estimating the required number of
resources, time to containment, and the resulting fire area were compared. Five examples
illustrate some of the computational differences between the simple and the complex methods. The equations for the two methods can be
used and programmed to estimate fire size and
perimeter, time-to-containment, and the number of resources required as a function of
spread rate.
Retrieval Terms:fire control, fire suppression,
planning, fire containment, dispatching
n ellipse of a specified length-towidth ratio can be used to estimate
fire size and shape for suppression
modeling purposes and also for field
use. 1,2A4-to-I length-to-width ratio was
suggested for grass fires3 and a 2-to-1
ratio for a forest fire burning under a
canopy.4 Mathematical and geometric
model~5~6
have been developed to examine the sensitivity of final fire size and
time-to-containmenl to suppression
force allocations. The models have been
extended to the elliptical model with a
constant rate of fireline construction. 7
Given a variable rate of fireline construction and an elliptical fire growth model,
either of two models-simple or cornplex-can be used to estimate the required number of resources, the time-tocontainment, and the resulting fire area.
In the complex model, the area and
perimeter of the expanding ellipse are
computed at multiples of some time
increment (specified by the user) and
partial containment at time ti contributes to the possible complete containment at time ti+l.A typical fire at times ti,
tic,, ti+%tic3 will look like an ellipse that
continually changes @g. IA). The complex containment model takes full ad-
vantage of the smaller size of the ellipse
at time ti compared with the sizes at later
times. Two computer simulation modelsFOCUS and FEES-use this method in
their initial attack evaluations. FOCUS
(Fire Operational Characteristics Using
Simulation) evaluates alternative fire
suppression programs at the Forest or
District level,8and FEES (Fire Economics Evaluation System) is being developed to evaluate fire program options
for nonsite specific areas.9
A simpler technique determines the
containment time tc such that the total
length of line produced at time t, equals
or exceeds the fire perimeter at time tc.
The line built up to time tc is compared
with the perimeter of the ellipse at time t,
and line built around the smaller perimeter at earlier time points is not considered @g. 1B). This model will be referred to as the simple containment
model.
This note describes the formulation
and development of the procedures used
in both the simple and complex containment models using an ellipse of a
given length-to-width ratio. Five tables
illustrate the difference between the two
models in terms of resulting fire areas,
does not spread beyond its contained
perimeter.
"I A list of resources is available for
each fire, and each resource must be
described in terms of its type (handcrew,
engine, bulldozer, helitack, smokejumpers, tractor-plow), arrival time (mea_suredfrom the reported time of the fire),
and production rate (chains pee hour).
8. Resource units are used as they
arrive on the fire and their total line
production is computed as the product
of total time on the fire by their production rate.
MATHEMATICALDEVELOPMENT
Complex Containment Model
Figure I-(A) In the complex containment
model, fireline construction starts a t t i m e ti
and continues along the perimeter of the
ellipse evaluated at times ti+l, ti+2, and ti+3
specified by the user. (B) In the simple con-
tainment model, times ti, ti+,, ti+*,ti+3 represent arrival times of resources, and I,represents time to control. At that time, fireline
computed equals or exceeds the fire perimeter.
number of resources used, and time
required to contain each fire. Given
resource arrivals, assumed pr-oductio
rates, and a range of spread rates, outputs of both computer simulation models indicate that variations in the timeto-containment and the number of resources used may lead to substantial
differences in suppression cost estimates.
time required for containment or when
the fire exceeds 100 acres.
3. The rate of fireline construction is
resource-specific and is constant for
each resource during the time required
for containment or when the fire exceeds
100 acres.
4. Airtankers are not used because of
the resulting complexity in fire shape
and the difficulty of interpreting their
effectiveness.
5. I n the complex containment
model, fireline construction proceeds on
both sides of the ellipse and may be applied at unequal rates. For example, a
five-penson handcrew may be split into a
two-person and a three-person crew
working on opposite flanks of the ellipse. The complex model can use headattack computations if the forward
spread rate is small enough relative to
the line construction rate from the available resources.$ Head attack begins at
the head of the fire and the resources
may be applied at unequal rates on both
sides of the ellipse.
6. All suppression work is assumed
to be 100 percent effective, and a fire
ASSUMPTIONS
Some simplifying assumptions were
made to calculate the number of required resources, area, and time-to-containment for each fire, given a forward
spread rate. Unless otherwise indicated,
all assumptions apply to both containment models:
I. The shape of a free-burning fire is
an ellipse with constant length-to-width
ratio. The point of origin is defined as
one of the focal points. Initial area atthe
time each fire is reported is one-tenth of
an acre (0.04 ha).
2. The forward rate-of-spread (at the
head of the fire) is constant during the
The complex containment model assumes an elliptical fire such that the
spread rate V from any point on the
perimeter at polar angle 4, measured
from the direction of most rapid spread
'$rh(t),is given by fig. 2)
in which e is the eccentricity of the fire
ellipse
Equa"lon I can be integrated with
respect to time along constant 4 to give
perimeter distance R Iirrom the origin of
the fire at time t:
in which h(t) is the distance of the fire
head from the origin, given by
A fire perimeter location y can be
expressed in rectangular coordinates
[x(t), y(x(t), t)] in which
and
Figure 2-In the complex model. line construclion is calculated separately for lower
and upper halves of the ellipse and can proceed toward the rear or head of the fire.
The forward rate-of-spread Vh(t) is
assumed to be constant and is matched
against a variable line construc"kon rate
VL(~)).Equation 4 can be integrated
numerically to calculak eonstrraetlon
progress as a function of h. The sign in
equation 4 is positive if progress is
toward the head of the ellipse; the sign is
negative otherwise. lo
Equations 2 and 3 are used to compute the (x, y) position of progress and
are calculated independently for the
lower and upper halves of "Le ellipse as a
function of the variable cons"cuction
rates for each half. Head or rear attack is
chosen on the basis of a specified ratio of
rate-of-spread to rate-of-available-lineconstruction. Unlike the simple containment model, equation 4 is evaluated and
integrated starting at the first arrival
time and thereafter at I-minute increments. Therefore, the available line construction rate is applied almost continu-ously as the ellipse expands with time.
Simple Containment Model
and
is an auxiliary variable.
Starting at time to and a perimeter
point (x(t,), y[x(to)]), control line construction can proceed toward the rear or
the head of the fire. Let the ratio of line
construction rate VL(~)to the forward
spread rate of the head Vh(t) be P at the
time the fire is at h(t). The position vector R(4, t) and its derivative with respect to time are (lig. 2)
In the simple containment10 model
the elliptical fire starts at time to = 0.0
k2 -4- (yx * k +- y)2
V2(t)
and is reported at time tr@g. 3). The fire
is contained at time t cif the perimeter at
Solving for ri.
time t c is less than or equal to the line
- y x y k J ~ Z ( t ) * ( ~ + y x ) 2 - y 2 built around the fire at time tc, i.e.,
x. = - dx
=
dt
1 4- Y?
k
C(E)* Vh * tc P (tc) = C ~ i ( t -c ti)
i= I
-dx
- - -y, y 5 d ~ t ( t *) (1
dt
(1
.
-
+
+ yX)2~x2)
*
h'(t)
- .
,,
y 2
-yxyh 2 J P ~* (I + yx)2- yh2
I + yx2
in which P = K ( t ) / Vh(t).
The derivatives of y with respect to x
and h written in terms of A, h, and z,
lead to
in which P(tc) is the perimeter at time t,,
and k is the number of resources dispatched to generate the required perimeter at time t, The corresponding area at
time tc will be
( ~ *h tc)2 * r
*
JI -
t2/
( I + e)2.
solving for tcgives:
in which a dot indicates the partial
derivative with respect to time and yx is
the partial derivative of y with respect to
X.
Setting the line construction rate VL(~) in which
dR
h(t) EZ
at time t equal to -gives
P2> A2 * dt
h(t) - tz
in which
Ri = line production rate of the ith
resource (chains per minute)
k
R2
Rk
3
Time
Figure 3-Atimesequenceexplainsvariables
in the simple model: a fire started attime tois
t, = time the fire was reported (minutes)
7 ,= ti - t, = travel time for the ith
resource (minutes)
'Vh = forward rate of spread (chains
per minute)
E ( ~ 2 ) is an elliptical integral, which
exists only in tabular form. The linear
function 5.83 - 3.766 is used as an approximation for C ( E ) in the range of
interest. C ( E ) is a unitless quantity.
The number of resources dispatched k
is determined by yk< t c < yk + I, or k
is equal to the maximum number of
resources available.
reported at time t, and controlled attime h;
resources arriveat time tiiwith production rate
Ri and travel time yi to the fire site.
into a computer program that calculated
area at containment, time to containment, and number of resources used, at
13 spread rates. The programs were used
to evaluate five different examples,
which varied in available resources, type
of attack, and length-to-width ratio of
the fire. Iln all cases, dispatch procedures
and spread rates were the same.
In the first case, the following five
resources were available for each fire:
sources already on the fire the resource
was not used. The length-to-width ratios
for the ellipse were 4: 1 and 2: 1. The contained areas and time to containment
did not differ substantially between the
two models for spread rates less than 10
chains per hour (eable I).
In the second case, the hllowing five
resources were available for all fires:
Time of
arrival (min)
TYpe
Assumed
production rate
23
41
50
Handcrew
(2 persons)
Handcrew
(2 persons)
Engine crew
(2 persons, 200 gal)
Engine crew
(2 persons, 200 gal)
Wandcrew
(2 persons)
40
6.8
45
6.8
45
16
69
16
6.8
ANALYSIS AND RESULTS
Two of the handcrews were the same. If
the arrival time of any resource exceeded
the required containment time by re-
Tv~e
Assumed
~roductionrate
Chains/ hour
15
Chainslhotrr
23
52
The simple and complex containment
model equations were each incorporated
Time of
arrival (min)
Engine crew
(2 persons, 200 gal)
Bulldozer
medium size
Handcrew
(2 persons)
Handcrew
(2 persons)
Engine crew
(2 persons, 200 gal)
.
I6
36
6.8
6.8
16
The length-to-width ratios for the ellipse
were 4: 1 and 2: 1. Due to the high production of the bulldozer, the complex
containment model used head attack
calculations up to a spread rate of 10
chains per hour (see assun~ption5). This
attack option resulted in a reduction of
acreage and number of crews rased at the
lower spread rates for the complex
model (table 2).
The third case was the same as the first
case. except that the complex model
used the head attack option. With the
head attack option, area a t containment
and the number of resources used differed substantially (table 3). At 5 chains
per hour, the complex model shifted
over to rear attack, and the results were
almost identical for both models. Again,
spread rates in excess of I0 chains per
hour were needed to show any differences in contained area.
In the fourth and fifth cases, the
resources were the same as for the
second case, but production rates differed. For the fourth case they were onefourth of those in the second case, and
for the fifth case they were one-half of
those in the second case:
Time of
a"ival(min)
Assu~ned
production rates
Fourth Fifth
case
case
Type
Chaim/hour
15
Engine crew
(2 ~ersons.200 "
gall
Bulldozer
~
40
x
1
(medium)
45
45
69
4
8
9
18
CONCLUSIONS
Handcrew
(2 persons)
Handcrew
(2 persons)
1.7
3.4
1.7
3.4
Engine crew
(2 persons, 200 gal)
4
8
Length-to-width ratios were 2: 1. In
the fourth case, area-at-containment
varied with increasing spread rates (table
4). For a spread of 8 chains per hour or
more, the crews cannot match the peri-
Table 1 -Area at csontainm~nt,rime to c'oiziainrnent,anclreso~~rc~r~c
~wecl,h13~preaclrateandImgrh-ton~itlthratio, fir sir?~p/e
and (40177pIe5 ~~01ztaii?t>7er~t
~IOLI~I,I
Area at
co~ltainlment
Spread
rate
(ch/hr)
Simple
A
1
~netergrowth rate in the simple model.
For the complex model, the fire escapes
(exceeds 100 acres) at I I chains per hour.
Again in the fifth case, for spread
rates in excess of 5 chains per hour, "ce
complex model gave smaller areas at
containment (table 5).
Time to
containment
Area ratio
Siinple
(sirnp1e:coinplex)
Complex
6.rc.s - -h f i r z ~ rt~~
4:l length-to-widtia ratio
Complex
%:I length-to-width ratio
Resources used
Simple
Complex
No empirical data are available to
verify the ,accuracy of the simple and
complex containment models. The results are rather limited in scope, but they
do illustrate some of the computational
differences between the two models. A
large number of possible combinations
of dispatches, fire spread rates, and line
construction rates would further bring
out differences between these two models. Variations in the time-to-containment and the number of resources used
may lead to substantial differences in
suppression cost estimated by the two
models.
The results indicate that:
The models agree closely for elliptical fires in which the ratio of line production rate to perimeter increase rate is
high.
The use of the direct head attack
option in the complex model significantly affects area at containment and
the number of resources used.
@ For those fires in which the production rate is close to the perimeter growth
rate, the complex model gives substantially smaller containment areas and
fewer escapes.
@ For the average fire the simple
model has an advantage of approximately 100 to 1 in computer running
time over the complex model. This
advantage increases as the time to containment goes up. The equations for the
simple model can easily be programmed
and adapted within other models.
Potential users of these models must
consider these results, computational
cost, ease of using the simple model, and
the range of production rate versus
perimeter growth rate, in deciding whether to use the simple or complex model.
@
@
END NOTES AND REFERENCES
'Anderson, Hal E. Predicting wind-driven wildland fire size and shape. Res. Paper INT-305.
Ogden, UT: Intermountain Forest and Range
Experiment Station, Forest Service, U.S. Department of Agriculture; 1983. 26 p.
ZGreen, D. 6.;Gill, A. M.;
Noble, I. R. Fire
&apes and the adequacy offire spread models.
Ecol. Model. 20: 33-45; 1983 March.
'Cheney, N. P.;Bary, G. A. V. Thepropagalion
ofn~assconflagration in a standing euca!tptforest
by the spolting process. In: Proceedings of the
1969 mass fire symposium; 1969 February 10-12;
Canberra, Australia. Maribyrnong, Victoria, Australia: Defense Standards Laboratory; 1969.
4Van Wagner, C. E. A simplefie-growth model.
For. Chron. 4(2): 103-104; 1969.
Sparks, G. M.; Jewell, W. S. A preliminary
model for initial attack. Res. Rep. F-1. Berkeley,
CA: Operation Research Center, University of
California; 1962. 29 p.
bAlbini, I". A.; Korovin, 6 . N.;
Gorovaya, E. PI.
Mathematical analj~siso m r e s t $re suppression.
Res. Paper INT-207. Ogden, UT: Intermountain
Forest and Range Experiment Station, Forest
Service, U.S. Department of Agriculture; 1978.
19 p.
7Albini, Frank A.; Chase, Carolyn M. Fire containment equations for pocket calculators. Res.
Note INT-268. Ogden, UT: Intermountain Forest
and Range Experiment Station, Forest Service,
U.S. Department of Agriculture; 1980. 17 p.
8Bratten, Frederick W.; Davis, James B.;
Flatman, George T.; Keith, Jerold W.; Rapp,
Stanley R.; Storey, Theodore 4;. FOCUS: Afire
management planning system-fial report. Gen.
Tech. Rep. PSW-49. Berkeley, CA: Pacific Southwest Forest and Range Experiment Station, Forest
Service, U.S. Department of Agriculture; 1981.
34 p.
gMills, Thomas J.; Bratten, Frederick W. FEES:
design of a Fire Economics Evaluation System.
Gen. Tech. Rep. PSW-65. Berkeley, CA: Pacific
Southwest Forest and Range Experiment Station,
Forest Service, U.S. Department of Agriculture;
1982. 26 p.
")The equations for the complex and simple
containment models were developed by Frederick
W, Bratten, operations research analyst, Pacific
Southwest Forest and Range Experiment Station,
Riverside, Calif.
andlength-toTable 2-Area at contlain~nenl,lime to confainmenl,andresources used, bj~spreadra'clte
\vidrh ratio, for simple and complex con lair amen^ tnodel'cls
4:1 length-to-width ratio
I
2
3
4
5
6
7
8
9
10
12
14
I6
0.16
.26
.39
.55
.77
1.04
1.36
1.78
2.32
2.94
4.8 1
7.15
1 1.03
0.14
.20
.27
.36
-46
.60
.73
.88
1.11
1.33
2.98
4.12
5.92
38
41
43
45
47
49
51
54
56
59
66
72
79
34
40
41
43
44
45
46
47
50
52
52
46
71
1.16
1.30
1.44
1.52
1.67
1.73
1.86
2.02
2.09
2.21
1.6 1
1.73
1.86
L
2
2
4
4
4
4
4
4
4
4
4
5
1
2
2
2
2
4
4
4
4
4
4
4
5
21 length-to-width ratio
1
2
3
4
5
6
7
8
9
10
12
14
16
.18
.36
.60
.9 1
1.34
1.80
2.52
3.60
4.89
6.58
11.27
18.12
28.98
.I5
.25
.36
.54
.73
.97
1.30
1.65
2.09
2.52
6.04
8.85
12.90
33
40
43
45
48
50
51
56
59
63
71
79
89
33
38
41
42
48
45
47
49
52
57
6.5
70
76
1.20
1.44
1.66
1.68
1.83
1.85
1.93
2.18
2.35
2.61
I .86
2.04
2.25
1
2
2
4
4
4
4
4
4
4
5
5
5
1
I
2
2
2
4
4
4
4
4
4
5
5
Table 4-Area af containment, time to co~ztainment,and resources used, by spread ratefor simple and
complex containment models. Length-to-width ratio of ellipse was 2:l
containment
Time to
containment
Area ratio
Resources used
Table 3--Area at contain~?zent,
i'irne to cor~tninment,a~zdresozrrce~
used
width ratio, for sirrzple and complex contain~lzenlmodels
I
Spread
rate
(ch/hr)
Simple
Complex
Acres
I
I
t
Area at
containment
I?,,~yreadraleandlengtlt-to-
Time to
containment
Simple
Complex
Area ratio
(simp1e:complex)
Resources used
Simple
Complex
--Minutes4:l length-to-width ratio
2:1 length-to-width ratio
Table 5-Area at containment, time to containment, and resources used, by spreadratefor simple and
complex containrnent models. Lennfh-to-widfh ratio o f ellipse was 2:l
Spread
rate
(ch/hr)
Area at
containment
Simple
Complex
Time to
containment
Simple
Complex
Area ratio
(simp1e:complex)
Resources used
Simple
Complex
A c r e s-
The Author:
ROMAIN M. MEES is a mathematician with the Station's unit studying fire management
planning and economics, in Riverside, California. He earned bachelor's and master's
degrees in mathematics at the University of California, Riverside. He joined the Station's
staff in 1971.
The Forest Service, U,S. Department of Agriculture, is responsible for Federal leadership in
forestry. It c d e s out this role through four main activities:
e Protection and management of resources om 191million acres of National Forest System lands.
*' Cooperation with State and Bocal governments, forest industries, and private landowners to
help protect and manage non-Federal forest and associated range and watershed lands.
e Pa~icipationwith other agencies in human resource and con~munltyassistance programs to
improve living conditions in rural areas.
Research on all aspects of forestv, rangeland management, and forest resources utilization.
The Pacific Southwest Forest and Range Experiment Station
* Represents the rcesewch branch of the Forest Service in California, Hawaii, and the western
Pacific.